I'm very new to stochastic calculus and I'm puzzled by what exactly is the difference between solving a stochastic differential equation vs solving a differential equation with stochastic initial conditions.
Most of the information I found online focused on the first kind of problem, i.e. we have a system described by differential equations and a stochastic source, e.g.
$$ \hat L [x(t)] = f(t),$$
where $x(t)$ is a variable of interest, $\hat L$ is some differential operator and $f(t)$ is a random noise term.
On the other hand, often if physics, we are interested in a case where a system is described by an ordinary differential equation, but the initial conditions are not known but can be described using a random variable. E.g. we can have
$$\hat L[x(t)] = 0, $$
and where $x_0 = x(t=0)$ is say a Gaussian random variable.
Are these two examples related? Can one system be mapped mathematically into another? Or are these completely separate type of problems?
A stochastic DE has - in the most simple setup where the filtration is generated by a Brownian motion - a whole Brownian path $ω$ as parameter. That is, $X_t(ω)=X(t,ω)$ where $ω$ is a continuous function and because of this simple setup the Brownian path to the stochastic variable $ω$ is $W_t(ω)=ω(t)$. These then get combined into the symbolism $$ dX_t(ω)=a(t,ω)dt+b(t,ω)dW_t(ω). $$ As you see in the infinitesimal increment of the Brownian path, which is almost surely not differentiable, this has no easy translation into an ordinary differential equation. You need the Ito or similar integral and stochastic calculus to ascribe meaning to this symbolism.